# zbMATH — the first resource for mathematics

Discrete invariants of varieties in positive characteristic. (English) Zbl 1084.14023
Let $$F: X \to S$$ be a smooth proper morphism of sheaves in characteristic $$p >0$$. If the sheaves $$R^b f_* \Omega^a_{X/S}$$ are locally free and the Hodge-de Rham spectral sequence degenerates at $$E_1$$, then the de Rham cohomology sheaves $$\mathcal M = H^m_{\text{dR}} (X/S)$$ are locally free $$\mathcal O_S$$-modules equipped with a descending Hodge filtration $$C^\bullet$$ and an ascending conjugate filtration $$D_\bullet$$ as well as with $$\mathcal O_S$$-linear isomorphisms $$\varphi_i: (\text{gr}^i_C)^{(p)} \to \text{gr}_i^D$$ given by the inverse Cartier operator. In this paper the authors give a complete classification of the structures $$(\mathcal M, C^\bullet, D_\bullet, \varphi_\bullet)$$ over an algebraically closed field. The result shows that such structures are essentially combinatorial objects.

##### MSC:
 14F40 de Rham cohomology and algebraic geometry 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 14G35 Modular and Shimura varieties 14J28 $$K3$$ surfaces and Enriques surfaces
##### Keywords:
Hodge structures; Cartier operator; de Rham cohomology
Full Text: